Problem: Simplify the following expression: $ t = \dfrac{q + 9}{-q + 7} - \dfrac{4}{7} $
Solution: In order to subtract expressions, they must have a common denominator. Multiply the first expression by $\dfrac{7}{7}$ $ \dfrac{q + 9}{-q + 7} \times \dfrac{7}{7} = \dfrac{7q + 63}{-7q + 49} $ Multiply the second expression by $\dfrac{-q + 7}{-q + 7}$ $ \dfrac{4}{7} \times \dfrac{-q + 7}{-q + 7} = \dfrac{-4q + 28}{-7q + 49} $ Therefore $ t = \dfrac{7q + 63}{-7q + 49} - \dfrac{-4q + 28}{-7q + 49} $ Now the expressions have the same denominator we can simply subtract the numerators: $t = \dfrac{7q + 63 - (-4q + 28) }{-7q + 49} $ Distribute the negative sign: $t = \dfrac{7q + 63 + 4q - 28}{-7q + 49}$ $t = \dfrac{11q + 35}{-7q + 49}$ Simplify the expression by dividing the numerator and denominator by -1: $t = \dfrac{-11q - 35}{7q - 49}$